A solution \((\tau , x)\)  of the problem (REF ) is called a \(P\) -solution of the Hamiltonian systems. The problem (REF ) has relation with the closed geodesics on Riemannian manifold (cf.[1]}) and symmetric periodic solution or the quasi-periodic solution problem (cf.[2]}). In addition, the first author C. Liu in [3]} transformed some periodic boundary problem for nonlinear delay differential systems and some nonlinear delay Hamiltonian systems to \(P\) -boundary problems of Hamiltonian systems as above, we also refer [4]}, [5]}, [6]}, [7]} and references therein for the background of \(P\) -boundary problems in \(N\) -body problems.
